In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M.

The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the same as a volume form. For the trivial group, an {e}-structure consists of an absolute parallelism of the manifold.

Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal ${\displaystyle G}$-bundle over a group ${\displaystyle G}$ "comes from" a subgroup ${\displaystyle H}$ of ${\displaystyle G}$. This is called reduction of the structure group (to ${\displaystyle H}$).

Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are G-structures with an additional integrability condition.

One can ask if a principal ${\displaystyle G}$-bundle over a group ${\displaystyle G}$ "comes from" a subgroup ${\displaystyle H}$ of ${\displaystyle G}$. This is called reduction of the structure group (to ${\displaystyle H}$), and makes sense for any map ${\displaystyle H\to G}$, which need not be an inclusion map (despite the terminology).

In the following, let ${\displaystyle X}$ be a topological space, ${\displaystyle G,H}$ topological groups and a group homomorphism ${\displaystyle \phi \colon H\to G}$.

Given a principal ${\displaystyle G}$-bundle ${\displaystyle P}$ over ${\displaystyle X}$, a reduction of the structure group (from ${\displaystyle G}$ to ${\displaystyle H}$) is a ${\displaystyle H}$-bundle ${\displaystyle Q}$ and an isomorphism ${\displaystyle \phi _{Q}\colon Q\times _{H}G\to P}$ of the associated bundle to the original bundle.

Given a map ${\displaystyle \pi \colon X\to BG}$, where ${\displaystyle BG}$ is the classifying space for ${\displaystyle G}$-bundles, a reduction of the structure group is a map ${\displaystyle \pi _{Q}\colon X\to BH}$ and a homotopy ${\displaystyle \phi _{Q}\colon B\phi \circ \pi _{Q}\to \pi }$.